Phonon-mediated room-temperature quantum Hall transport in graphene

The quantum Hall (QH) effect in two-dimensional electron systems (2DESs) is conventionally observed at liquid-helium temperatures, where lattice vibrations are strongly suppressed and bulk carrier scattering is dominated by disorder. However, due to large Landau level (LL) separation (~2000 K at B = 30 T), graphene can support the QH effect up to room temperature (RT), concomitant with a non-negligible population of acoustic phonons with a wave-vector commensurate to the inverse electronic magnetic length. Here, we demonstrate that graphene encapsulated in hexagonal boron nitride (hBN) realizes a novel transport regime, where dissipation in the QH phase is governed predominantly by electron-phonon scattering. Investigating thermally-activated transport at filling factor 2 up to RT in an ensemble of back-gated devices, we show that the high B-field behaviour correlates with their zero B-field transport mobility. By this means, we extend the well-accepted notion of phonon-limited resistivity in ultra-clean graphene to a hitherto unexplored high-field realm.

2 Supplementary Note 1: Temperature dependence of the phonon occupation at high fields According to Ref. [24], the inverse of the electronic magnetic length determines the wave-vectors of phonons contributing to electron-phonon scatterings in magnetic fields. As a consequence, the energy scale ℎ = ℏv ⁄ is introduced. Moreover, due to the high energy range of optical phonons (1800-2300 K) leading to small phonon occupation numbers, only acoustic phonons can be considered. In Figure S1, we therefore show the field dependence of Eph (inset), as well as the phonon occupation number at B = 30 T (main panel), for the longitudinal and transverse acoustic phonons in graphene (both included in the calculations of Ref. [24]).

Supplementary Note 2: Zero-field temperature-dependent resistivity
In Figure S2, we show the zero-field resistivity of devices D1-3, measured in temperature and carrier density ranges relevant to our study (T = 170 K -300 K, n = 1 × 10 12 cm -2 ). For all samples, a clear monotonic increase of the resistivity can be observed, as generally accepted for high-mobility single-layer graphene and attributed to carrier scattering with thermally excited phonons [5][6][7][8][9][10]. From linear fits to the data, we find slopes within 0.24 /K and 0.45 /K, comparable with Ref.
[5] (~0.3 /K at the same carrier density, obtained from the linear fit shown in Figure S2). In accordance with the mobility data discussed in the main text (Figure 2a), we observe that device D3 closely approximates the behaviour of reference data from Ref.
[5], indicating analogous sample quality, and dominant e-ph scattering close to RT. The resistivity of devices D1 and D2, as expected from their lower carrier mobility, is offset due to residual disorder scattering adding to the e-ph mechanism.

Supplementary Note 3: Low-temperature transport at high magnetic fields
As well-established in hBN-encapsulated graphene devices, the application of high magnetic fields promotes correlation-driven phases. Accordingly, both integer QH states outside the half-integer 2, 6, 10 ,… sequence, and fractional QH states are observed at low temperature in our samples (data for sample D2 are shown in Figure S3; see Ref. [31] for sample D4). 5 Supplementary Note 4: Phonon-mediated limit at ν = -2 Given the particle-hole symmetric electronic structure of graphene, the same behaviour observed at filling factor ν = 2 (see main text) should be followed by the activated resistivity at ν = -2. In Figure S4 we show that this is indeed the case: data from samples D1-D3 group close to the e-ph limit, with D3 reasonably following the exact theoretical curve (see inset). 6

Supplementary Note 5: Arrhenius plots
Given the 1/T-exponential dependence of the activated conductivity, it is not surprising that an Arrhenius-type behaviour can describe our data. When considering the pre-factor σArr ( Figure S5e), the clear contrasting behaviour between disordered and clean samples can be appreciated. While data from Ref.
[20] approximate the long-range disorder limit (2e 2 /h multiplied by the factor 4, accounting for spin and valley degeneracy), our samples show dramatically lower values. As discussed in the manuscript, the magnitude of the pre-factor for sample D3 matches the predictions of Ref.
[24], while small corrections need to be considered for D1 and D2, correlating with their lower zero-field mobility. at different magnetic fields. The activation energy is kept at half of the bare LL gap, following the results of Ref.
[20]. b, Arrhenius fits for sample D2. c, Arrhenius fits for sample D3. d, Arrhenius fits on data from Ref. [20]. e, Conductivity pre-factors σArr from the fits in panels a-d. The shaded areas correspond to ± standard error on the best-fit intercept from linear fits to ln(σxx) vs 1/T.

Supplementary Note 6: Characteristic field and temperature dependence of the e-ph pre-factor
In this section, we investigate the temperature and field dependence of the e-ph pre-factor form data on samples D1-3. Taking into account the term σ0 from Ref.
[24] and the constant correction σD (expressed as resistivity ρD in Figure 4 and relative discussion), the activated conductivity of our samples can be expressed as σxx = (σ0 + σD ) exp(-ΔLL/2kT) = (σN σT σB + σD)exp(-ΔLL/2kT) , where we name σT and σB the temperature and field dependent components, respectively. From our data, we can easily extract σT and σB for each temperature and magnetic field, and compare them with the predictions of Ref.
However, we stress that the exponential term exp(-ΔLL/2kT) depends both on temperature and field (due to the B-dependence of the LL gap ΔLL). As shown in the Figure S6a, within 180 K and 300 K, the exponential results in a relative increase of the conductivity exceeding by more than one order of magnitude the predicted σT. As a consequence, the T-linear contribution is likely to remain elusive. Within 20 T and 30 T, instead, σB results in a relative variation of the conductivity in the same order of the exponential term ( Figure S6d). Therefore, the B 1/2 dependence might be more realistically observed.
In Figure S6b we show the obtained σT as a function of T (with data points averaged over the different magnetic fields), compared to σT = T/300 K (dark cyan line). No temperature dependence can be conclusively identified from our data, in agreement with the scenario anticipated above. We note that similar fluctuations in the data points are also observed in the pre-factor to the data from Ref.
[20] ( Figure S6c). Hence, we conclude that the exponential term precludes the experimental assessment of a possible temperature dependence in the conductivity pre-factor.
In Figure S6e we show that σB (averaged over the different temperatures) follows the expected B 1/2 dependence. The quantitative agreement with the field dependence proposed in Ref.
[24] supports the conclusions of our manuscript. In contrast, data from Ref.
[20] are in accordance with a constant conductivity pre-factor ( Figure S6f).

Supplementary Note 7: Temperature-driven interplay between disorder and phonon contributions
To gain a deeper insight on the interplay between disorder and phonon scattering in clean samples, we perform additional experiments at lower magnetic fields (produced by a standard superconducting coil) and, correspondingly, lower temperatures. We fabricate an additional device (D5), following the same methods of D1-3 (exfoliated graphene flakes). D5 shows transport characteristics fully comparable to D3 (at 220 K, we obtain n* ~ 4 x 10 10 cm -2 and carrier mobility ~1.4 x 10 5 cm 2 V -1 s -1 ), indicating that it is representative of the clean limit discussed in the main text.
We collect data at B ≥ 1 T, to ensure full development of the QH effect at υ = 2, as well as an activation energy matching ΔLL/2. At B = 1 T, we employ temperatures of 50 K to 100 K to observe activated behaviour (see Figure   S7, panel a). According to Ref. [24], at T = 100 K e-ph scattering should result in a resistivity of ~100 Ω. Despite the high sample quality of D5, we measure values larger by more than one order of magnitude (~1.4 kΩ). This observation contrasts with the high-B high-T behaviour, where negligible deviation from the e-ph limit is measured for sample D3. By increasing the applied magnetic field and, accordingly, the temperature range, the activated resistivity drops, and the deviation from the e-ph limit tends to be suppressed (see panels b, c and d in Figure S7). We quantify the deviation at different magnetic fields and temperatures using the parameter ρD introduced in the main text, obtained by fits shown as black lines in Figure S7. lines are fits to the activated resistivity, employed to extract the parameter ρD shown in Figure S8. The inset in panel d shows the raw resistivity data as a function of the filling factor at B = 10 T.
A more comprehensive picture can be proposed by combining the ρD parameters extracted for two samples of comparable quality, D5 and D3, as shown in Figure S8a. At B = 10 T, ρD drops by approximately a factor 2 with respect to 1 T. At B ≥ 20 T, ρD vanishes (these are the same data points for D3 shown in main text Figure 4). The corresponding temperature ranges employed in the measurements are shown in Figure S8b. Throughout the experiments, the temperature ranges are adapted to the applied magnetic field, to follow the increase of the gap size and ensure that all the data are collected in comparable conditions of thermal activation. Overall, our results indicate that high-quality graphene devices can show both disorder and phonon-mediated contributions to the dissipative conductivity in the QH regime. The disorder contribution is non-universal, meaning that it is sample-dependent and tends to be suppressed with increasing temperature (and magnetic field). In clean enough samples, the e-ph contribution becomes dominant toward RT, where the QH effect is observable only at high magnetic fields (B ≥ 20 T).